JEE Mains · Maths · STD 12 - 7.2 definite integral
The of value the integral \(\frac{48}{\pi^{4}} \int_{0}^{\pi}\left(\frac{3 \pi x ^{2}}{2}- x^{3}\right) \frac{\sin x }{1+\cos ^{2} x } dx\) is equal to
- A \(6\)
- B \(7\)
- C \(8\)
- D \(9\)
Answer & Solution
Correct Answer
(A) \(6\)
Step-by-step Solution
Detailed explanation
\(I=\frac{48}{\pi^{4}} \int_{0}^{\pi} x^{2}\left(\frac{3 \pi}{2}-x\right) \frac{\sin x}{1+\cos ^{2} x} d x\) Apply king property \(I=\frac{48}{\pi^{4}} \int_{0}^{\pi}(\pi-x)^{2}\left(\frac{\pi}{2}+x\right) \frac{\sin x}{1+\cos ^{2} x} d x\) \((1) + (2)\)…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- The value of \(\left(2 .{ }^{1} P _{0}-3 .{ }^{2} P _{1}+4 .{ }^{3} P _{2}-\ldots .\right.\) up to \(51\) th term)+\(\left(1 !-2 !+3 !-\ldots . .\right.\) up to \(51^{\text {th }}\) term \()\) is equal toJEE Mains 2020 Medium
- If \(\sum_{\mathrm{r}=1}^9\left(\frac{\mathrm{r}+3}{2^{\mathrm{r}}}\right) .{ }^9 \mathrm{C}_{\mathrm{r}}=\alpha\left(\frac{3}{2}\right)^9-\beta, \quad \alpha, \beta \in \mathrm{N}, \quad\) then \((\alpha+\beta)^2\) is equal toJEE Mains 2025 Medium
- Which of the following points lies on the locus of the foot of perpendicular drawn upon any tangent to the ellipse, \(\frac{x^{2}}{4}+\frac{y^{2}}{2}=1\) from any of its foci?JEE Mains 2020 Hard
- If the mean and variance of eight numbers \(3,7,9,12,13,20, x\) and \(y\) be \(10\) and \(25\) respectively, then \(\mathrm{x} \cdot \mathrm{y}\) is equal toJEE Mains 2020 Hard
- The cost of running a bus from \(A\) to \(B\), is \(Rs.\,\left( {av + \frac{b}{v}} \right)\). where \(v\, km/ h\) is the average speed of the bus. When the bus travels at \(30\, km/h\), the cost comes out to be \(Rs.\, 75\) while at \(40\, km/h\), it is \(Rs.\,65\) . Then the most economical speed (in \(km/ h\)) of the bus isJEE Mains 2013 Hard
- If the sum of the first four terms of an A.P. is 6 and the sum of its first six terms is 4, then the sum of its first twelve terms isJEE Mains 2026 Hard
More PYQs from JEE Mains
- The positive value of the determinant of the matrix \(A\), whose \(A d j(A d j(A))=\left(\begin{array}{ccc}14 & 28 & -14 \\ -14 & 14 & 28 \\ 28 & -14 & 14\end{array}\right)\), isJEE Mains 2022 Hard
- Let \(f(x)=\begin{cases} e^{x-1}, & x<0 \\ x^2-5x+6, & x \geq 0 \end{cases}\) and \(g(x)=f(|x|)+|f(x)|\). If the number of points where \(g\) is not continuous and is not differentiable are \(\alpha\) and \(\beta\) respectively, then \(\alpha+\beta\) is equal to ______JEE Mains 2026 Hard
- If the point \(P\) on the curve, \(4 x^{2}+5 y^{2}=20\) is farthest from the point \(Q (0,-4),\) then \(PQ ^{2}\) is equal toJEE Mains 2020 Medium
- The area (in \(sq. \,units\)) of the region, given by the set \(\left\{(x, y) \in R \times R \mid x \geq 0,2 x^{2} \leq y \leq 4-2 x\right\}\) is:JEE Mains 2021 Hard
- If \([x]\) is the greatest integer \(\leq x\), then \(\pi^{2} \int_{0}^{2}\left(\sin \frac{\pi \mathrm{x}}{2}\right)(\mathrm{x}-[\mathrm{x}])^{[\mathrm{x}]} \mathrm{d} \mathrm{x}\) is equal to :JEE Mains 2021 Hard
- Let \(\displaystyle\int_{-2}^{2} (|\sin x| + [x \sin x])\,dx = 2(3 - \cos 2) + \beta\), where \([\cdot]\) is the greatest integer function. Then \(\beta \sin\left(\dfrac{\beta}{2}\right)\) equals:JEE Mains 2026 Medium